Encyclopedia of Systems and Control

2015 Edition
| Editors: John Baillieul, Tariq Samad

Bounds on Estimation

  • Arye Nehorai
  • Gongguo Tang
Reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5058-9_69

Abstract

We review several universal lower bounds on statistical estimation, including deterministic bounds on unbiased estimators such as Cramér-Rao bound and Barankin-type bound, as well as Bayesian bounds such as Ziv-Zakai bound. We present explicit forms of these bounds, illustrate their usage for parameter estimation in Gaussian additive noise, and compare their tightness.

Keywords

Barankin-type bound Cramér-Rao bound Mean-squared error Statistical estimation Ziv-Zakai bound 
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Notes

Acknowledgements

This work was supported in part by NSF Grants CCF-1014908 and CCF-0963742, ONR Grant N000141310050, AFOSR Grant FA9550-11-1-0210.

Bibliography

  1. Barankin EW (1949) Locally best unbiased estimates. Ann Math Stat 20(4):477–501MathSciNetGoogle Scholar
  2. Bell KL, Steinberg Y, Ephraim Y, Van Trees HL (1997) Extended Ziv-Zakai lower bound for vector parameter estimation. IEEE Trans Inf Theory 43(2):624–637Google Scholar
  3. D’Andrea AN, Mengali U, Reggiannini R (1994) The modified Cramér-Rao bound and its application to synchronization problems. IEEE Trans Commun 42(234):1391–1399Google Scholar
  4. Forster P, Larzabal P (2002) On lower bounds for deterministic parameter estimation. In: IEEE international conference on acoustics, speech, and signal processing (ICASSP), 2002, Orlando, vol 2. IEEE, pp II–1141Google Scholar
  5. Gorman JD, Hero AO (1990) Lower bounds for parametric estimation with constraints. IEEE Trans Inf Theory 36(6):1285–1301MathSciNetGoogle Scholar
  6. Hochwald B, Nehorai A (1994) Concentrated Cramér-Rao bound expressions. IEEE Trans Inf Theory 40(2): 363–371MathSciNetGoogle Scholar
  7. Hochwald B, Nehorai A (1997) On identifiability and information-regularity in parametrized normal distributions. Circuits Syst Signal Process 16(1):83–89MathSciNetGoogle Scholar
  8. Kay SM (2001a) Fundamentals of statistical signal processing, volume 1: estimation theory. Prentice Hall, Upper Saddle River, NJGoogle Scholar
  9. Kay SM (2001b) Fundamentals of statistical signal processing, volume 2: detection theory. Prentice Hall, Upper Saddle River, NJGoogle Scholar
  10. Marzetta TL (1993) A simple derivation of the constrained multiple parameter Cramér-Rao bound. IEEE Trans Signal Process 41(6):2247–2249Google Scholar
  11. Rockah Y, Schultheiss PM (1987) Array shape calibration using sources in unknown locations – part I: far-field sources. IEEE Trans Acoust Speech Signal Process 35(3):286–299Google Scholar
  12. Stoica P, Nehorai A (1989) MUSIC, maximum likelihood, and Cramér-Rao bound. IEEE Trans Acoust Speech Signal Process 37(5):720–741MathSciNetGoogle Scholar
  13. Stoica P, Ng BC (1998) On the Cramér-Rao bound under parametric constraints. IEEE Signal Process Lett 5(7):177–179Google Scholar
  14. Tichavsky P, Muravchik CH, Nehorai A (1998) Posterior Cramér-Rao bounds for discrete-time nonlinear filtering. IEEE Trans Signal Process 46(5):1386–1396Google Scholar
  15. Van Trees HL (2001) Detection, estimation, and modulation theory: part 1, detection, estimation, and linear modulation theory. Jhon Wiley & Sons, Hoboken, NJGoogle Scholar

Copyright information

© Springer-Verlag London 2015

Authors and Affiliations

  • Arye Nehorai
    • 1
  • Gongguo Tang
    • 2
  1. 1.Preston M. Green Department of Electrical and Systems EngineeringWashington University in St. LouisSt. LouisUSA
  2. 2.Department of Electrical Engineering & Computer ScienceColorado School of MinesGoldenUSA